Goat tied to a tree

[Korlek]

Hi,
I have a problem with one exercise from Analysis. Here it is: A goat is tied to a a tree by a rope. Both radius of a tree and length of a rope are 1. The quetion is: what's the area of grass, that goat can eat?
I was trying to solve it by my own, but I can't. It's of course half circle and two parts, which unfortuantelly aren't part of a circle. I tried to find a function that describe that curve, but I failed.
Do you have any ideas how to do it?

 

[Quaid]

[the area of the grass that the goat can eat is] of course half circle and two parts, which unfortuantelly aren't part of a circle.

I think those two parts are actually part of the same circle. Think about overlapping circles.



The circle on the left is the tree trunk; the circle on the right is how the roped goat could move, if the tree was not there. The green region is the area you need to find.

Does this picture give you any ideas about how to set up an area function?

Cheers :cool:

 

[pka]

I have a problem with one exercise from Analysis. Here it is: A goat is tied to a a tree by a rope. Both radius of a tree and length of a rope are 1. The quetion is: what's the area of grass, that goat can eat?
I was trying to solve it by my own, but I can't. It's of course half circle and two parts, which unfortunately aren't part of a circle. I tried to find a function that describe that curve, but I failed. Do you have any ideas how to do it?

This question has a long history. Look at this discussion.
In this problem has a difference. The tree has radius 1. The rope has length 1.
That means that the goat can wrap around the tree an arc length in each direction.

At the bottom of that page are more links.